Finite Group Databasebeta

← #30650690

Subgroups — #30650690

Order 16256.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550529C211yesmaximal, characteristic, complemented
2#22550519C211yesmaximal, characteristic, complemented
3#22550205C211yesmaximal, characteristic, complemented
4#22550193C211yesmaximal, characteristic, complemented
5#22550163C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550088C211yesmaximal, characteristic, complemented
9#22550024C211yesmaximal, characteristic, complemented
10#22550011C211yesmaximal, characteristic, complemented
11#22549989C211yesmaximal, characteristic, complemented
12#22549949C211yesmaximal, characteristic, complemented
13#22549876C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#489×C127C211yesmaximal, characteristic, complemented
16C2×#14523249—21no—
17C2×#14523249C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19C2×#14523235C2211yescharacteristic, complemented
20C2×D1016C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22C2×#14523231C2211yescharacteristic, complemented
23C2×C8×D127C2211yescharacteristic
24#16861194—21no—
25#16861194—21no—
26#16861191—21no—
27#16861185—21no—
28#16861185C2211yescharacteristic
29#16861184C2211yescharacteristic
30#16861184C2211yescharacteristic, complemented
31#16861182—21no—
32#16861180C2211yescharacteristic
33#16861179—21no—
34C4×D508C2211yescharacteristic
35C4×D508C2211yescharacteristic
36C4×D508C2211yescharacteristic, complemented
37C4⋊C4×D127C2211yescharacteristic
38C4⋊C4×D127C2211yescharacteristic, complemented
39C22⋊C4×D127—21no—
40#16861141C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42#16861140C2211yescharacteristic, complemented
43C4⋊(C127⋊C8)C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52C4⋊Q8×C127C2211yescharacteristic, complemented
53#124×C127C2211yescharacteristic
54#103×C127C2211yescharacteristic, complemented
55#103×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523249—22no—
61#14523249—41no—
62#14523249—44no—
63C2×D508—21no—
64C2×D508—21no—
65C2×D508—21no—
66C2×D508D411yescharacteristic
67C2×D508C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71D4×D127—22no—
72D4×D127—41no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127C2311yescharacteristic
79C2×C4×D127C2311yescharacteristic
80#14523235—41no—
81D1016—22no—
82#14523233—41no—
83#14523231—41no—
84C2×C127⋊C8C2311yescharacteristic
85C2×C127⋊C8C2311yescharacteristic
86C8×D127—41no—
87C4⋊Dic127D411yescharacteristic
88C4⋊Dic127C2311yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92Dic127⋊C4—21no—
93C22⋊Dic127—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4D411yescharacteristic
100(C2×D127)⋊C4D411yescharacteristic
101C2×Q8×C127—21no—
102C2×Q8×C127C2311yescharacteristic
103C2×D4×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127—21no—
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C22⋊C4×C127—21no—
111C42×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—42no—
115C22×D127—42no—
116C22×D127—81no—
117C22×D127C2×D411yescharacteristic
118C22×D127C2×D411yescharacteristic
119D508—22no—
120D508—21no—
121D508—21no—
122D508—21no—
123D508—42no—
124D508—42no—
125D508—42no—
126D508#4112yescomplemented
127C22⋊D127—42no—
128C22⋊D127—81no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—21no—
137C4×D127—42no—
138C4×D127—42no—
139C4×D127—42no—
140C4×D127—42no—
141C4×D127—81no—
142C127⋊C8—21no—
143C127⋊C8—21no—
144C23×C127—21no—
145Q8×C127—21no—
146Q8×C127—21no—
147Q8×C127—42no—
148D4×C127—21no—
149D4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—21no—
165C2×D127—22no—
166C2×D127—21no—
167C2×D127—21no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—84no—
174C2×D127AGL(1,ℤ/8)12yes—
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127—41no—
180C4×C127C2×#4111yescharacteristic
181C4×C127H(5,𝔽2)11yescharacteristic
182Dic127—21no—
183Dic127—21no—
184Dic127—41no—
185Dic127—41no—
186C2×C127—81no—
187C2×C127#54611yescharacteristic
188C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C127#57711yescharacteristic
190D127—22no—
191D127—41no—
192D127—42no—
193D127—81no—
194#3306—1271nomaximal, Sylow
195C127#330611yescharacteristic, complemented, Sylow
196#547—1271no—
197#546—1271no—
198#489D12711yescharacteristic, complemented
199#483—1271no—
200#481—1271no—
201#476—1271no—
202#469—1271no—
203#465—1271no—
204#463—1271no—
205#449—1271no—
206#447—1271no—
207#442—1271no—
208#435—1271no—
209#417—1271no—
210C2×#102—1271no—
211C2×#41—1271no—
212C2×#41—2541no—
213C22×D4—1271no—
214C2×SD16—1271no—
215C2×D8—1271no—
216C2×D8—1271no—
217C2×M16—1271no—
218C22×C8—1271no—
219C4⋊Q8C2×D12711yescharacteristic, complemented
220#124C2×D12711yescharacteristic
221#123—2541no—
222#123—2541no—
223#123—2541no—
224C22⋊Q8—1271no—
225C22⋊Q8—2541no—
226#121—1271no—
227#121—1271no—
228#121—1271no—
229#121—2541no—
230#120—2541no—
231C4×D4—1271no—
232C4×D4—1271no—
233C4×D4—1271no—
234C2×C4⋊C4—1271no—
235C2×C4⋊C4—1271no—
236C2×C22⋊C4—2541no—
237#106—1271no—
238#106—1271no—
239#106—1271no—
240C4⋊C8—1271no—
241#103—1271no—
242#103C2×D12711yescharacteristic, complemented
243#103C2×D12711yescharacteristic, complemented
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102C2×D12711yescharacteristic
250#102C2×D12711yescharacteristic
251C22⋊C8—1271no—
252C22⋊C8—1271no—
253#97C2×D12711yescharacteristic, complemented
254C24—2541no—
255#41—5084no—
256#41—5081no—
257#41—2542no—
258C2×Q8—21no—
259C2×Q8C22×D12711yescharacteristic
260C2×D4—1271no—
261C2×D4—1271no—
262C2×D4—1271no—
263C2×D4—5081no—
264C2×D4—2541no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2542no—
269C2×D4—2541no—
270C2×D4C22×D12711yescharacteristic
271C22×C4—1271no—
272C22×C4—1271no—
273C22×C4—2541no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4—2541no—
277C22×C4—2541no—
278SD16—5081no—
279D8—5081no—
280D8—2542no—
281M16—5081no—
282C2×C8—1271no—
283C2×C8—1271no—
284C2×C8—5081no—
285C2×C8C22×D12711yescharacteristic
286C2×C8C22×D12711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—1271no—
289C4⋊C4—1271no—
290C4⋊C4—1271no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4—2541no—
294C4⋊C4C22×D12711yescharacteristic
295C4⋊C4C22×D12711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—1271no—
299C22⋊C4—1271no—
300C22⋊C4—2541no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C42C22×D12711yescharacteristic
307C23—21no—
308C23—1271no—
309C23—1271no—
310C23—10161no—
311C23—5082no—
312C23—5082no—
313C23—2541no—
314C23—2541no—
315Q8—21no—
316Q8—21no—
317Q8—42no—
318D4—21no—
319D4—21no—
320D4—1272no—
321D4—10161no—
322D4—5082no—
323D4—5082no—
324D4—5082no—
325D4—5082no—
326D4—2542no—
327D4—2541no—
328D4—2541no—
329D4—2541no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—1271no—
334C2×C4—1271no—
335C2×C4—10161no—
336C2×C4—5082no—
337C2×C4—5082no—
338C2×C4—5082no—
339C2×C4—5082no—
340C2×C4—2541no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4D4×D12711yescharacteristic
347C2×C4D4×D12711yescharacteristic
348C2×C4C23×D12711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2541no—
352C8—2541no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—1272no—
357C22—10164no—
358C22—5081no—
359C22—5081no—
360C22—5081no—
361C22—5081no—
362C22—5081no—
363C22—2541no—
364C22—2541no—
365C22—2542no—
366C22—2541no—
367C22—2541no—
368C22—2541no—
369C22C2×D4×D12711yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—5081no—
376C4—5081no—
377C4—2541no—
378C4—2541no—
379C4#41×D12711yescharacteristic
380C4#1686125511yescharacteristic
381C2—81no—
382C2—10161no—
383C2—5082no—
384C2—5081no—
385C2—2542no—
386C2#2255051911yescharacteristic
387C2AGL(1,ℤ/8)×D12711yescharacteristic
388C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#489×C127C2
socleC22×C127C22×D4
hypercenter#489D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550529C2semidirect
2. #22550519C2semidirect
3. #22550205C2semidirect
4. #22550193C2semidirect
5. #22550163C2semidirect
6. #22550128C2semidirect
7. #22550108C2semidirect
8. #22550088C2semidirect
9. #22550024C2semidirect
10. #22550011C2semidirect
11. #22549989C2semidirect
12. #22549949C2semidirect
13. #22549876C2semidirect
14. #102×D127C2semidirect
15. #489×C127C2semidirect
19. C2×#14523235C22semidirect
22. C2×#14523231C22semidirect
30. #16861184C22semidirect
36. C4×D508C22semidirect
38. C4⋊C4×D127C22semidirect
40. #16861141C22semidirect
41. #16861140C22semidirect
42. #16861140C22semidirect
43. C4⋊(C127⋊C8)C22semidirect
44. #16861133C22semidirect
46. #16861131C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. C4⋊Q8×C127C22semidirect
54. #103×C127C22semidirect
55. #103×C127C22semidirect
58. #97×C127C22semidirect
126. D508#41semidirect
195. C127#3306semidirect
198. #489D127semidirect
219. C4⋊Q8C2×D127semidirect
242. #103C2×D127semidirect
243. #103C2×D127semidirect
253. #97C2×D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225505292
#225505192
#225502052
#225501932
#225501632
#225501282
#225501082
#225500882
#225500242
#225500112
#225499892
#225499492
#225498762
#102×D1272
#489×C1272
#3306127